Simplify the following expression and state the condition under which the simplification is valid: $q = \dfrac{p^2 + 9p + 18}{p^2 + 6p}$
First factor the expressions in the numerator and denominator. $ \dfrac{p^2 + 9p + 18}{p^2 + 6p} = \dfrac{(p + 3)(p + 6)}{(p)(p + 6)} $ Notice that the term $(p + 6)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(p + 6)$ gives: $q = \dfrac{p + 3}{p}$ Since we divided by $(p + 6)$, $p \neq -6$. $q = \dfrac{p + 3}{p}; \space p \neq -6$